Find the gradient of $f(x, y, z) = xy + yz + zx$ at $(-4, 3, 1)$. $\nabla f(-4, 3, 1) = ($
Answer: The gradient of a scalar field is all its partial derivatives put together into a vector. For a 3D scalar field, this looks like $\nabla f = (f_x, f_y, f_z)$. Let's find $f_x$, $f_y$, and $f_z$. $\begin{aligned} f_x &= \dfrac{\partial}{\partial x} \left[ xy + yz + zx \right] \\ \\ &= y + z \\ \\ f_y &= \dfrac{\partial}{\partial y} \left[ xy + yz + zx \right] \\ \\ &= x + z \\ \\ f_z &= \dfrac{\partial}{\partial z} \left[ xy + yz + zx \right] \\ \\ &= y + x \end{aligned}$ Now we can evaluate the partial derivatives we found at the point $(-4, 3, 1)$. $\begin{aligned} f_x(-4, 3, 1) &= y + z = 4 \\ \\ f_y(-4, 3, 1) &= x + z = -3 \\ \\ f_z(-4, 3, 1) &= y + x = -1 \end{aligned}$ The gradient of $f$ at $(-4, 3, 1)$ is $\nabla f = (4, -3, -1)$.